#112257

order	512
nilpotency class	3
isoclinism	isoclinic class of #112245
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,7,4,9)(3,11,14,8,13,16,12,15)(17,18)(19,29)(20,34,38,44)(21,27)(22,26)(23,28)(24,31)(25,37,43,36)(30,45,47,39)(32,41)(33,40)(35,42,46,48);(2,7)(3,12)(4,5)(8,16)(11,15)(13,14)(17,19,23,33)(18,24,28,41)(20,35,38,46)(21,32,22,31)(25,42,43,48)(26,40,27,29)(30,37,47,36)(34,39,44,45);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,34)(27,44)(28,43)(29,42)(31,35)(32,46)(33,47)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19453);